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<h1>Minimal phase spectral factorization</h1>
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<pre class="codeinput">
<span class="comment">% A PSD matrix is found which minimizes a weighted trace while obtaining</span>
<span class="comment">% fixed sums along the diagonals. Notice the use of a FOR loop to access</span>
<span class="comment">% the diagonals of X. A later version of CVX will eliminate the need for</span>
<span class="comment">% this by allowing the use of the SPDIAGS function in side models.</span>
<span class="comment">% Nevertheless, this example provides an illustration of the use of</span>
<span class="comment">% standard Matlab control statements to build models.</span>
<span class="comment">%</span>
<span class="comment">% Adapted from an example provided in the SeDuMi documentation.</span>

<span class="comment">% Generate data</span>
b = [2; 0.2; -0.3];
n = length( b );

<span class="comment">% Create and solve model</span>
cvx_begin <span class="string">sdp</span>
    variable <span class="string">X( n, n )</span> <span class="string">symmetric</span>
    dual <span class="string">variable</span> <span class="string">y{n}</span>
    dual <span class="string">variable</span> <span class="string">Z</span>
    minimize( ( n - 1 : -1 : 0 ) * diag( X ) );
    <span class="keyword">for</span> k = 1 : n,
        sum( diag( X, k - 1 ) ) == b( k ) : y{k};
    <span class="keyword">end</span>
    X &gt;= 0 : Z;
cvx_end
y = [ y{:} ]';

<span class="comment">% Display resuls</span>
disp( <span class="string">'The optimal point, X:'</span> );
disp( X )
disp( <span class="string">'The diagonal sums:'</span> );
disp( sum( spdiags( X, 0:n-1 ), 1 ) );
disp( <span class="string">'min( eig( X ) ) (should be nonnegative):'</span> );
disp( min( eig( X ) ) )
disp( <span class="string">'The optimal weighted trace:'</span> );
disp( ( n - 1 : -1 : 0 ) * diag( X ) );
</pre>
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<pre class="codeoutput">
 
Calling SDPT3 4.0: 6 variables, 3 equality constraints
   For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------

 num. of constraints =  3
 dim. of sdp    var  =  3,   num. of sdp  blk  =  1
*******************************************************************
   SDPT3: Infeasible path-following algorithms
*******************************************************************
 version  predcorr  gam  expon  scale_data
   HKM      1      0.000   1        0    
it pstep dstep pinfeas dinfeas  gap      prim-obj      dual-obj    cputime
-------------------------------------------------------------------
 0|0.000|0.000|6.9e-01|5.3e+00|3.0e+02| 2.000000e+01  0.000000e+00| 0:0:00| chol  1  1 
 1|1.000|1.000|2.5e-07|5.7e-02|1.9e+01| 1.882061e+01  2.306667e+00| 0:0:00| chol  1  1 
 2|0.921|1.000|1.4e-07|5.7e-03|1.5e+00| 4.040616e+00  2.524843e+00| 0:0:00| chol  1  1 
 3|1.000|0.859|8.8e-08|1.3e-03|3.7e-01| 4.105341e+00  3.748230e+00| 0:0:00| chol  1  1 
 4|0.983|0.985|1.9e-08|7.5e-05|6.0e-03| 3.880804e+00  3.875203e+00| 0:0:00| chol  1  1 
 5|0.988|0.988|4.9e-10|6.4e-06|7.0e-05| 3.877308e+00  3.877275e+00| 0:0:00| chol  1  1 
 6|0.986|0.988|6.1e-12|7.7e-08|9.2e-07| 3.877268e+00  3.877268e+00| 0:0:00| chol  1  1 
 7|1.000|1.000|3.3e-14|1.2e-12|3.7e-08| 3.877267e+00  3.877267e+00| 0:0:00|
  stop: max(relative gap, infeasibilities) &lt; 1.49e-08
-------------------------------------------------------------------
 number of iterations   =  7
 primal objective value =  3.87726745e+00
 dual   objective value =  3.87726741e+00
 gap := trace(XZ)       = 3.73e-08
 relative gap           = 4.26e-09
 actual relative gap    = 4.26e-09
 rel. primal infeas (scaled problem)   = 3.29e-14
 rel. dual     "        "       "      = 1.21e-12
 rel. primal infeas (unscaled problem) = 0.00e+00
 rel. dual     "        "       "      = 0.00e+00
 norm(X), norm(y), norm(Z) = 2.4e+00, 1.9e+00, 2.0e+00
 norm(A), norm(b), norm(C) = 3.8e+00, 3.2e+00, 3.1e+00
 Total CPU time (secs)  = 0.10  
 CPU time per iteration = 0.01  
 termination code       =  0
 DIMACS: 3.5e-14  0.0e+00  1.2e-12  0.0e+00  4.3e-09  4.3e-09
-------------------------------------------------------------------
 
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.122733
 
The optimal point, X:
    0.0468   -0.0369   -0.3000
   -0.0369    0.0292    0.2369
   -0.3000    0.2369    1.9240

The diagonal sums:
    2.0000    0.2000   -0.3000

min( eig( X ) ) (should be nonnegative):
   5.7391e-09

The optimal weighted trace:
    0.1227

</pre>
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